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Computer Science and Engineering
ComputingANear-MaximumIndependentSetinLinearTimebyReducing-Peeling
Lijun Chang
University of New South Wales, [email protected]
Joint work with Wei Li, Wenjie Zhang
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Introduction
Given a graph ! = ($, &), a vertex subset ( ⊆ $ is an independent set if for any two vertices * and + in (, there is no edge between *and + in !.
Independent Set
An independent set(of ! is a maximum independent set if its size is the largest among all independent sets of !.
Maximum Independent Set
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Introduction
Given a graph ! = ($, &), a vertex subset ( ⊆ $ is an independent set if for any two vertices * and + in (, there is no edge between *and + in !.
Independent Set
An independent set(of ! is a maximum independent set if its size is the largest among all independent sets of !.sets of !.
Maximum Independent Set
Independent Set
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Introduction
Given a graph ! = ($, &), a vertex subset ( ⊆ $ is an independent set if for any two vertices * and + in (, there is no edge between *and + in !.
Independent Set
An independent set(of ! is a maximum independent set if its size is the largest among all independent sets of !.
Maximum Independent Set
MaximumIndependent
Set
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Introduction
Applicationsv Build index for shortest path/distance queries [Cheng et al.
SIGMOD’12, Fu et al. VLDB’13]v Refine the result of matching two graphs [Zhu et al. VLDB J’13]v Social network coverage [Puthal et al. BigData’15]; vertex
cover
Hardnessv NP-hard to compute a maximum independent set [Garey et al.
Book’79]v Hard to approximate
§ NP-hard to approximate within a factor of -./0 for any 0 <3 < 1 [J. Håstad. FOCS’96]
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Existing WorksExact algorithms -- branch-and-reduce paradigm
v [F. V. Fomin et al .J.ACM’09] § Theoretically runs in 5∗ 1.22019 time
v [T. Akiba et al. Theor. Comput. Sci.’16]§ Practically computes the exact solution for many small and
medium-sized graphs
Approximation algorithmsv [U. Feige J. Discrete Math’04, M. M. Halldórsson et al.
Algorithmica’97, P. Berman. Theor.Comput. Sys.’99]§ Approximation ratio largely depends on n or Δ§ Not practically useful
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Existing Works
Heuristic algorithms for large graphsv Linear-time algorithms
§ Greedy, dynamic update§ Efficient, but can only find small independent sets in
practicev Iterative randomized searching
§ Local search algorithm: ARW [D. V. Andrade. J.Heuristics’12]
§ Evolutionary algorithm: ReduMIS [S. Lamm. ALENEX’16]§ Local search + simple reduction rules: OnlineMIS [J.
Dahlum. SEA’16]§ Can find large independent sets, but take long time
Our goal: find large independent sets in a time-efficientand space-effective manner
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Three Observations Utilized in Our Frameworkv Observation–I: Real networks are usually power-law graphs with
many low-degree vertices
v Observation-II: Reduction rules have been effectively used for low-degree vertices
v Observation-III: High-degree vertices are less likely to be in a maximum independent set
;< =>? = @ ∝B
@C
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Three Observations Utilized in Our Frameworkv Observation–I: Real networks are usually power-law graphs with
many low-degree vertices
v Observation-II: Reduction rules have been effectively used for low-degree vertices
(b) Isolation D ! = D !\ +, F(c) Folding D ! = D !/ *, +, F + 1
v Observation-III: High-degree vertices are less likely to be in a maximum independent set
(a) D ! = D !\ +
Degree-one Reduction
Degree-two ReductionsD ! : J-KLML-KL-NL-*OPLQRS!
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Three Observations Utilized in Our Frameworkv Observation–I: Real networks are usually power-law graphs with
many low-degree vertices
v Observation-II: Reduction rules have been effectively used for low-degree vertices
v Observation-III: High-degree vertices are less likely to be in a maximum independent set
Ø If a high-degree vertex is added into the independent set, then all its neighbors, which are of a large quantity, are ruled out from the independent set [J. Dahlum et al SEA’16]
Ø Removing/peeling high-degree vertices can further sparsify the graph [Y. Lim et al TKDE’14]
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The Reducing-Peeling FrameworkDefinition 3.1: (Inexact Reduction) Given a graph !, we remove/peel the vertex with the highest degree from !.
v Phase 1: ReducingØ While a reduction rule can be applied on a vertex * then
Apply the exact reduction rule on *
v Phase 2: PeelingØ Apply the inexact reduction rule to temporarily remove a high-
degree vertex
v Repeat the above two phases until there is no edge in the graph
v Post-process: Iteratively add a temporarily removed vertex to the solution if the independence requirement is not violated
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Overview of Our Approaches
v Compute large independent set for large graphs in a time-efficient and space-effective manner§ Subquadratic (or even linear) time.§ 2m + O(n) space: m is the number of undirected edges.
§ A graph is stored in 2m + n + O(1) space by the adjacency array (aka, Compressed Sparse Row) graph representation
§ A graph with one billion edges takes slightly more than 8GBmemory
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An Efficient Baseline Algorithmv BDOne
Step 1:While $T. ≠ ∅ or $WX ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily
removed vertices
=>? YB = B
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An Efficient Baseline Algorithmv BDOne =>? YB = B
YZ[\][^_^_>_[?_>\^=>?<>>
Step 1:While $T. ≠ ∅ or $WX ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily
removed vertices
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An Efficient Baseline Algorithmv BDOne =>? YB = B
YZ[\][^_^_>_[?_>\^=>?<>>
Step 1:While $T. ≠ ∅ or $WX ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily
removed vertices
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An Efficient Baseline Algorithmv BDOne =>? YB = B
YZ[\][^_^_>_[?_>\^=>?<>>
Step 1:While $T. ≠ ∅ or $WX ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily
removed vertices
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An Efficient Baseline Algorithmv BDOne =>? YB = B
YZ[\][^__[?_>\^=>?<>>
Complexity AnalysisTime: 5 OSpace: 2O + 5 -
Step 1:While $T. ≠ ∅ or $WX ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily
removed vertices
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An Effective Baseline Algorithmv BDTwo
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwo-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily removed
vertices
=>? YB = B
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An Effective Baseline Algorithmv BDTwo =>? YB = B
=>? Ya = b
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwo-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily removed
vertices
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An Effective Baseline Algorithmv BDTwo =>? YB = B
=>? Ya = b
Complexity AnalysisTime: 5 -×O andg O + -hRi-Space: 6O + 5 -
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwo-Reduction
ElseInexact-Reduction
Step 2:Recover temporarily removed
vertices
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An Effective Linear-Time Algorithmv LinearTime
Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., +X, … , +m , let + ∉ k and F ∉ k be the unique vertices connected to +. and +m, respectively.
Case 1: + = F
⟹ D ! = D !\ +
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An Effective Linear-Time Algorithmv LinearTime
Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., +X, … , +m , let + ∉ k and F ∉ k be the unique vertices connected to +. and +m, respectively.
Case 2: k is odd and +, F ∈ &
⟹ D ! = D !\ +, F
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An Effective Linear-Time Algorithmv LinearTime
Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., +X, … , +m , let + ∉ k and F ∉ k be the unique vertices connected to +. and +m, respectively.
Case 3: k is odd and +, F ∉ &
⟹ D !
= D !\ +X, … , +m ⋃ +., F +k − 1
2
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An Effective Linear-Time Algorithmv LinearTime
Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., +X, … , +m , let + ∉ k and F ∉ k be the unique vertices connected to +. and +m, respectively.
Case 4: k is even and +, F ∈ &
⟹ D !
= D !\ +., … , +m +k
2
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An Effective Linear-Time Algorithmv LinearTime
Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., +X, … , +m , let + ∉ k and F ∉ k be the unique vertices connected to +. and +m, respectively.
Case 5: k is even and +, F ∉ &
⟹ D != D !\ +., … , +m ⋃ +, F
+k
2
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
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An Effective Linear-Time Algorithmv LinearTime
Step 1:While $T. ≠ ∅ or $TX ≠ ∅ or $W` ≠ ∅
If $T. ≠ ∅ thenDegreeOne-Reduction
Else if $TX ≠ ∅ thenDegreeTwoPath-Reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
Complexity AnalysisTime: 5 OSpace: 2O + 5 -
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A Near-Linear-Time Algorithmv NearLinear
Lemma 5.1: (Dominance Reduction) [F. V. Fomin et al. JACM’09]Vertex v dominates vertex u if +, * ∈ & and all neighbors of v other than u are also connected to u (i.e., s + \ * ⊆ s * ). If v dominates u, then there exists a maximum independent set of G the excludes u; thus, we can remove u from G, and D ! = D !\ * .
Lemma 5.2: Vertex v dominates its neighbor u iff ∆ +, * = K + − 1, where ∆ +, * is the number of triangles containing u and v
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A Near-Linear-Time Algorithmv NearLinearStep1: Maintain the set u of candidate dominated vertices, and also maintain ∆ +, *for every edge +, *
Step 2:While $TX ≠ ∅ or u ≠ ∅ or $W` ≠ ∅
If $TX ≠ ∅ thenDegreeTwoPath-Reduction
Else if u ≠ ∅thendominance reduction
ElseInexact-Reduction
Step 2: Recover temporarily removed vertices
Complexity AnalysisTime: 5 O×Δ(Δ is the maximum degree in !)
Space: 4O + 5 - in worst case and 2O + 5 - in practice
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Extensions of Our Algorithms
v Accelerate ARW
v Compute Upper Bound of D !
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Experimental Settingsv Datasets
v EnvironmentsØ All algorithms are
implemented in C++Ø All experiments are
conducted on a machine with an Intel(R) Xeon(R) 3.4GHz CPU and 16GB main memory running Linux
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Accuracy
v Gap to the maximum independent set size
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Processing Time
(a) Compared with ExistingTechniques
(b) Compare Our Techniques
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Memory Usage(a) Compared
with ExistingTechniques
(b) Compare Our Techniques
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Boost ARW
ARW-NL, ARW-LT: ARW boosted by NearLinear and LinearTime, respectively.
Convergence plots of local search algorithms
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Outline
q Introduction
q Existing Works
q Our Reducing-Peeling Framework
q Our Approaches
q Experimental Studies
q Conclusion
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Conclusion
p A new Reducing-Peeling framework
p Time-efficient and space-effective techniques to implement the reducing-peeling framework
p Find large independent sets efficiently for large real-world graphs with billions of edges
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